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Lecture 4 Sequences

Lecture 4 Sequences. CSCI – 1900 Mathematics for Computer Science Fall 2014 Bill Pine. Lecture Introduction. Reading Rosen - Section 2.4 Sequences Definition Properties Recursion Characteristic Function. Sequence . A sequence is an ordered list of objects It can be finite

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Lecture 4 Sequences

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  1. Lecture 4Sequences CSCI – 1900 Mathematics for Computer Science Fall 2014 Bill Pine

  2. Lecture Introduction • Reading • Rosen - Section 2.4 • Sequences • Definition • Properties • Recursion • Characteristic Function CSCI 1900

  3. Sequence • A sequence is an ordered list of objects • It can be finite • 1,2,3,2,1,0,1,2,3,1,2 Or infinite • 3,1,4,1,5,9,2,6,5,… • In contrast to sets, with sequences • Duplicates are significant • Order is significant CSCI 1900

  4. Sequence Examples • 1, 2, 3, 2, 2, 3,1 is a sequence, but not a set • The sequences 1, 2, 3, 2, 2, 3, 1 and 2, 2, 1, 3 are made from elements of the set {1, 2, 3} • The sequences 1, 2, 3, 2, 2, 3, 1 and 2, 1, 3, 2, 2, 3, 1 only switch two elements, but that is sufficient to make them unequal CSCI 1900

  5. Alphabetic Sequences • Finite list of characters • D,i,s,c,r,e,t,e • d,a,b,d,c,a • Infinite lists of characters • a,b,a,b,a,b,a,b… • This,is,the,song,that,never,ends,It,goes,on,and,on,my,friends, Someone,started,singing,it,not,knowing,what,it,was,and,they’ll,continue,singing,it,forever,just,because… • String: a sequence of letters or symbols written without commas CSCI 1900

  6. Linear Array • Principles of sequences can be applied to computers, specifically, arrays • There are some differences though • Sequence • Well-defined • Modification of any element or its order results in a new sequence • Array • May not have all elements initialized • Modification of array by software may occur • Even if the array has variable length, we’re ultimately limited to finite length CSCI 1900

  7. Set Corresponding to a Sequence • Set corresponding to a sequence - the set of all distinct elements of the sequence • {a,b} is the set corresponding to sequence abababab… • {1,2,3} is the set corresponding to the sequences • 1,2,3 • 1,2,3,3,3,3,2,1,2,1 • 1,2,3,3,2,1,1,2,3,3,2,1…. CSCI 1900

  8. Defining Infinite Sequences • Explicit Formula, • The value for the nth item – an – is determined by a formula based solely on n • Recursive Formula • The value for the nth item is determined by a formula based on n and the previous value(s) – an , an-1 , … – in the sequence CSCI 1900

  9. Explicit Sequences • Easy to calculate any specific element • By convention a sequence starts with n = 1 • an = 2 * n • 2, 4, 6, 8, 10, 12, 14, 16, … • a3 = 2 * 3 = 6 • a5 = ??? • an = 3 * n + 6 • 9, 12, 15, 18, 21, 24, … • a5 = 3 * 5 + 6 = 21 • a100 = ??? • Defines the values a variable is assigned in a program loop when adding/subtracting/multiplying/dividing by a constant CSCI 1900

  10. for n = 1 thru 100 by 1 y = n * 5 display “y sub” n “is equal to” y next n Explicit Sequences and a Program yn = 5 *n in this sequence CSCI 1900

  11. Recursive Sequences • Used when the nth element depends on some calculation on the prior element(s) • You must specify the values of the first term(s) • Example: a1 = 1a2 = 2an = an-1 + an-2 • Defines the values a variable is assigned in a program loop when the current value depends upon its previous value(s) CSCI 1900

  12. yMinus2 = 1 print “y sub 1 is equal to” yMinus2 yMinus1 = 2 print “y sub 2 is equal to” yMinus1; for n = 3 thru 100 by 1 y = yMinus1 + yMinus2 print “y sub” n “ is equal to” y yMinus2 = yMinus1 yMinus1 = y next n Recursive Sequences and a Program yn = yn-1+yn-2 in this sequence CSCI 1900

  13. 1 if xA 0 if xA Characteristic Functions • Characteristic function – function defining membership in a set, over the universal set • fA(x) = • Example A = {1, 4, 6} Where U = Z+ • fA(1) = 1, fA(2) = 0, fA(4) = 1, fA(5) = 0, fA(6) = 1, fA(7) = 0, … • fA(0) = ??? CSCI 1900

  14. Representing Sets with a Computer • Recall: sets have no order and no duplicated elements • The need to assign each element in a set to a memory location • Gives the set order in a computer • We can use the characteristic function to define a set with a computer program CSCI 1900

  15. Characteristic Function in Programming • To see a simplistic implementation, consider the following example with A ={1,4,6} function isInA( x ) if( (x == 1) OR (x == 4) OR (x == 6) ) return ( 1 ) else return ( 0 ) • Might be used to validate user input CSCI 1900

  16. Properties of Characteristic Functions Characteristic functions of subsets satisfy the following properties (proofs are on page 15 of textbook) • fAB(x) = fA(x)fB(x) • fAB(x) = fA(x) + fB(x) – fA(x)fB(x) • fAB(x) = fA(x) + fB(x) – 2fA(x)fB(x) CSCI 1900

  17. Proving Characteristic Function Properties • A way to prove these is by enumeration of cases • Example: Prove fAB = fAfB fA(a) = 0, fB(a) = 0, fA(a) fB(a) = 0  0 = 0 = fAB(a)fA(b) = 0, fB(b) = 1, fA(b) fB(b) = 0  1 = 0 = fAB(b)fA(c) = 1, fB(c) = 0, fA(c) fB(c) = 1  0 = 0 = fAB(c)fA(d) = 1, fB(d) = 1, fA(d) fB(d) = 1  1 = 1 = fAB(d) a A B d b c CSCI 1900

  18. More Properties of Sequences • Any set with n elements can be arranged as a sequence of length n, but not vice versa • Sets have no order and duplicates don’t matter • Each subset can be identified with its characteristic function • A sequence of 1’s and 0’s • Characteristic function for universal set, U, is fU(x) = 1 CSCI 1900

  19. Countable and Uncountable • A set is countable if it corresponds to some sequence • Members of set can be arranged in a list • Elements have position • All finite sets are countable • Some, but not all infinite sets are countable, otherwise are said to be uncountable • An example of an uncountable set is set of real numbers • E.g., what comes after 1.23534 ? CSCI 1900

  20. Countable Sets • A finite set S is always countable • It can correspond the sequence • an = n for all n N where n  |S| • Countable { x | x = 5 * n and n Z+ } • Corresponds to the sequence an = n for all CSCI 1900

  21. Strings • Sequences can be made up of characters • Example: W, a, k, e, , u, p • Removing the commas and you get a string “Wake up” • Strings can illustrate the difference between sequences and sets more clearly • a, b, a, b, a, b, a, … is a sequence, i.e., “abababa…” is a string • The corresponding set is {a, b} CSCI 1900

  22. Key Concepts Summary • Sequences • Integers • Strings • Recursion • Characteristic Function CSCI 1900

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